Dynamics for the Fractional Nonlinear Schrodinger Equation

In this talk, we consider the dynamical properties of solutions to the
fractional
nonlinear Schrodinger equation (FNLS, for short) arising from
pseudorelativistic Boson stars. First, by establishing the profile decomposition
of bounded sequences in H^s, we find the best constant of a Gagliardo-Nirenberg
type inequality. Then, we obtain the stability and instability of standing waves
for (FNLS) by the profile decomposition. Finally, we investigate the dynamical
properties of blow-up solutions for (FNLS), including sharp threshold mass,
concentration and limiting profile. (Joint joint with Jian Zhang)